find all local max,min and/or saddle point of f(x,y)=4+x^3+y^-3xy

- anonymous

find all local max,min and/or saddle point of f(x,y)=4+x^3+y^-3xy

- katieb

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- anonymous

would you plz write the function using equation

- amistre64

hmm, something to do with:\[f_{xy}f_{yx}-(f_{xx})^2\]
or did i recall that incorrectly?

- amistre64

almost had it :)
\[f_{xx}f_{yy}-(f_{xy})^2\]

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## More answers

- anonymous

\[4+x ^{3}+y ^{3}-3xy\]

- anonymous

it fxx.fyy-(fxy)^2

- anonymous

i jst want to verify the answer for it !

- TuringTest

what answer did you get?

- anonymous

\[fx = 2x-3y \]
\[fy= 2y-3x\]
on solving both equations the point is (0,0)
\[fxx = 2\] , \[fyy= 2\] \[fxy = \]
\[D = -5 <0 saddle point\]

- TuringTest

that's not quite right I don't think

- anonymous

fxy = -3

- TuringTest

\[f(x,y)=4+x^3+y^3-3xy\]\[f_x=3x^2-3y=0\implies y=x^2\]\[f_y=3y^2-3x=3x^4-3x=3x^3(x-1)=0\]\[ x=\{0,1\},y=\{0,1\}\]

- anonymous

its x^3
so i have (0,0) ,(1,1)and a (-1,1)
i am not sure if (-1,1) should be there oor not

- anonymous

@Living_dreams I solved what you mentioned in your comment

- TuringTest

there is no x=-1 solution, I don't see how you get that

- TuringTest

@Avva I don't see how you get your fx and fy

- anonymous

Ooh I saw the powers on X ^ y 2 not 3

- anonymous

@Avva thanks for ur help but i was wonder abt the max and min points! n\ @TuringTest : 3y^2-3x=0 i sub y=x^2 from other equation so it will be 3x(x^3-1)=0
therefore you have x =0,1,-1
i am not sure abt -1 though !! tht was my question

- TuringTest

x^3=1 does not have x=-1 as a solution since (-1)^3=-1, not 1

- TuringTest

I did screw up my factoring earlier though, you were right about that :P

- TuringTest

3x(x^3-1)=0
3x=0 -> x=0
x^3=1 -> x=1

- anonymous

@TuringTest that alrite!! :)
so now my question is x^3=1 so will i hv x=-1 if i solve for x ?

- TuringTest

no, again, because \((-1)^3\neq1\)
so \(x^3=1\) has only one real solution, x=1
x=-1 is NOT a solution

- anonymous

the other two roots are complex not real

- anonymous

@TuringTest oh ok thanks a lot for ur help!!
so the final ans will be (0,0 ) is a saddle point and (1,1) will be a local minimum !

- TuringTest

exactly :)

- anonymous

@Avva :thanks for ur help !! :)

- anonymous

@Living_dreams URW :) sorry I didn't get that the powers are 3 at first

- anonymous

it okay !!i knw it get hard to tell the difference between 2 and 3 if we type it in equation form !!

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